A simple exponential relationship has this form:

Exponential graphs can be used to model or describe lots of practical situations.  Some of the common ones are growth and decline of a population of animals or humans, or how quickly water drains out of a hole at the bottom of a container.

However, to be able to model many of these situations, we need to make this simple general exponential relationship a bit more complicated.  First of all, we can add a coefficient:

We can also add or subtract another constant term from the ‘x’ in the power, like this:

And lastly, we can add or subtract another constant term from the whole thing:

So in this new general exponential relationship, ‘a’, ‘b’, ‘c’ and ‘d’ are all constants.  We can get back to our original simple form by making ‘b’ equal to one, and ‘c’ and ‘d’ equal to zero.

The graph we get when we set ‘a’ to ‘2’ say is this:

Changing ‘a’

Now, we can make ‘a’ bigger or smaller and see how that affects the graph:

Values of ‘a’ larger than 1 result in graphs that rise upwards as you go from left to right.  The larger ‘a’, the more steeply the graph rises up.  For values of ‘a’ smaller than 1, the graph slopes downwards going from left to right.  The smaller the value of ‘a’ below 1, the higher it will start on the left, and the more steeply it will slope down towards the x-axis before levelling out just above it to the right.

Changing ‘b’

Changing ‘b’ scales the graph in the vertical direction:

Compared to the ‘b = 1’ line (the middle one), the ‘b = 2’ line is twice as high above the x-axis along its entire length.  The ‘b = 0.5’ line is half as high above the x-axis as the ‘b = 1’ line.

It can get confusing telling the difference between graphs where ‘a’ has been changed, and ones where ‘b’ has been changed.  The trick is to look at where the lines cross the y-axis.  When ‘a’ is changed between different lines, they will still all pass through the same point on the y-axis.  When ‘b’ is changed for each of the lines however, each line will pass through a different point on the y-axis.

Changing ‘c’

Having a non-zero value of ‘c’ moves the line horizontally.  If ‘c’ is larger than zero, it moves the graph ‘c’ units to the left.  If ‘c’ is smaller than zero, it moves the graph ‘c’ units to the right.

See how the  graph is the  graph, but shifted one unit to the left all along the line.  Opposite goes for the  line.

Changing ‘d’

‘d’ is the easiest one to deal with.  If it is positive, the graph is shifted vertically upwards ‘d’ units.  If it is negative, it’s shifted downwards ‘d’ units.  Easy!

Notice how a vertical gap of 5 units looks a lot bigger when the slope of the two lines is almost flat.  When the lines are sloping steeply, it appears that they are closer together, but that’s because your eyes automatically have a tendency to look at the shortest distance between them, rather than the vertical distance between them.

### Negative powers

You can also change the graph by putting a negative sign in front of the ‘x’ in the power:

Doing this flips the graph horizontally across the y-axis:

Now how does varying the ‘c’ part now affect the graph?  Well, varying c moves the graph horizontally, just like when there was a positive sign in front of the ‘x’.  But which way?  If ‘c’ is positive, which way does the graph move?

To answer this, think about just .  As you move to the right of the graph, what happens to the overall power?  Well, it gets smaller.  Think about it – when x = –5, the power is +5.  When you move one unit to the right to x = –4, the power changes to +4 – it gets smaller.  Therefore as you move right on a line, the overall power gets smaller.

So say we start with the graph , and we want to produce the graph .  Let’s pick a point on our graph of  – say the point at x = –5.  At x = –5, the overall power in  is +5.  At x = –5 in our new graph, the overall power is +4.  So at x = –5 on our new line, we should plot the point that is at  on our old graph of .  In other words, we take the old graph and shift it one unit to the left.

### The discussion bit about populations

A lot of these questions will ask you to discuss how realistic the formulas are for population growth.  In this case you need to talk about the following things:

Deaths.  Whatever thing you’re talking about, it will eventually die.  Better models take this into account. Death comes to us all in the end…

Limited amount of sustainable food in the environment.  If a bunch of rabbits demolish all the green vegetation in an area, this is not sustainable – nothing will grow back and they’ll all starve to death.

Limited amount of shelter and breeding places.  This is especially important for animals like birds on rocky islands – there are only so many suitable places to nest.

Predators.  If the number of prey animals increases, the predators will do well and they will also increase in numbers and eat more and more of the prey animals.

Disease.  As the population gets larger and more crowded, there will be more chance for diseases and other similar things to take their toll.